A simple sub-quadratic algorithm for computing the subset partial order

A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been presented that compute the partial order (and thereby the minimal and maximal...

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Published in:Information processing letters Vol. 56; no. 6; pp. 337 - 341
Main Author: Pritchard, Paul
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 22.12.1995
Elsevier Science
Subjects:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Computer science; control theory; systems
Exact sciences and technology
Extremal sets
Lexicographic order
Mathematical logic, foundations, set theory
Mathematics
Sciences and techniques of general use
Set theory
Set-theoretic algorithms
Subset graph
Subset partial order
Theoretical computing
Set-theoretic algorithms
Extremal sets
Subset partial order
Algorithms
Lexicographic order
Subset graph
Subset relation
Partial ordering
Set theory
Algorithm complexity
ISSN:0020-0190, 1872-6119
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Abstract A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been presented that compute the partial order (and thereby the minimal and maximal sets, i.e., extremal sets) in worst-case time O( N 2 log N ) . This paper develops a simple algorithm that uses only simple data structures, and gives a simple analysis that establishes the above worst-case bound on its running time. The algorithm exploits a variation on lexicographic order that may be of independent interest.
AbstractList A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been presented that compute the partial order (and thereby the minimal and maximal sets, i.e., extremal sets) in worst-case time O( N 2 log N ) . This paper develops a simple algorithm that uses only simple data structures, and gives a simple analysis that establishes the above worst-case bound on its running time. The algorithm exploits a variation on lexicographic order that may be of independent interest.
Author Pritchard, Paul
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Cites_doi 10.1007/BF01261654
10.1137/S0097539791194094
10.1137/0216062
10.1016/0020-0190(93)90264-A
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Issue 6
Keywords Set-theoretic algorithms
Extremal sets
Subset partial order
Algorithms
Lexicographic order
Subset graph
Subset relation
Partial ordering
Set theory
Algorithm complexity
Language English
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References Paige, Tarjan (BIB3) 1987; 16
Pritchard (BIB6) March 1995
Dietzfelbinger, Karlin, Mehlhorn, der Heide, Rohnert, Tarjan (BIB2) 1994; 23
Pritchard (BIB7) March 1995
Pritchard (BIB8) 1995
Pritchard (BIB4) 1991; 28
Pritchard (BIB5) September 1994
Yellin, Jutla (BIB9) 1993; 48
Aho, Hopcroft, Ullman (BIB1) 1974
Aho (10.1016/0020-0190(95)00165-4_BIB1) 1974
Pritchard (10.1016/0020-0190(95)00165-4_BIB6) 1995
Pritchard (10.1016/0020-0190(95)00165-4_BIB5) 1994
Pritchard (10.1016/0020-0190(95)00165-4_BIB8) 1995
Yellin (10.1016/0020-0190(95)00165-4_BIB9) 1993; 48
Pritchard (10.1016/0020-0190(95)00165-4_BIB4) 1991; 28
Pritchard (10.1016/0020-0190(95)00165-4_BIB7) 1995
Dietzfelbinger (10.1016/0020-0190(95)00165-4_BIB2) 1994; 23
Paige (10.1016/0020-0190(95)00165-4_BIB3) 1987; 16
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  article-title: Finding extremal sets in less than quadratic time
  publication-title: Inform. Process. Lett.
  doi: 10.1016/0020-0190(93)90264-A
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SubjectTerms Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Computer science; control theory; systems
Exact sciences and technology
Extremal sets
Lexicographic order
Mathematical logic, foundations, set theory
Mathematics
Sciences and techniques of general use
Set theory
Set-theoretic algorithms
Subset graph
Subset partial order
Theoretical computing
Title A simple sub-quadratic algorithm for computing the subset partial order
URI https://dx.doi.org/10.1016/0020-0190(95)00165-4
Volume 56
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